Steel lathe tools that are sharp can do fantastic things, but once you've tried a carbide blade you'll become spoiled. Buying professional carbide tipped lathe tools can get expensive very fast. Fortunately you can make your own very cheaply.
I put this tool together using 1/2" steel square stock, some pine trim, a copper plumbing fitting, and a cutter bought on eBay. The cutter was $7 including shipping and came with a set screw. The rest of the materials, including a small amount of epoxy to hold it together can be found at any hardware store.
The cost for this cutter, including the drill bits and tap, was under $20. I have enough square stock, pine, and epoxy left over to make three more tools.
Basically, a square channel was cut in the pine trim with the router to hold the square stock. The actual cutter was screwed to the square stock after drilling and tapping the hole, and the rod was then inserted into the pine handle. The handle was turned on an old lathe using a scroll chuck and live tail stock but other methods could be devised to hold the handle for turning if a scroll chuck isn't available.
The video shows the build.
For mobile devices here is the link: https://www.youtube.com/watch?v=oohYDYjs8Pk